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2015 AMC 10B Problem 1

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2015 AMC 10B Problem 1AMC 10 Step by Step

Name: Date: September 13, 2026

Exponents, Logarithms & Radicals · difficulty 1 of 5 · about 1 min · position P1-10

What is the value of $2-(-2)^{-2}$ ?

A)
$-2$
B)
$\dfrac{1}{16}$
C)
$\dfrac{7}{4}$
D)
$\dfrac{9}{4}$
E)
$6$

Answer

Key insight

A negative exponent means reciprocal, and squaring kills the sign: (-2)^(-2) = 1/4, so the value is 2 - 1/4.

Solution

Handle the power first. A negative exponent is a reciprocal:
$(-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4},$
and the square of $-2$ is positive $4$ , so the sign disappears.

Then
$2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}.$

The answer is $\boxed{\textbf{(C)}\ \frac{7}{4}}$ .

Why this works

$a^{-n} = 1/a^n$ for any nonzero $a$ , and an even power of a negative base is positive. Order of operations matters: the exponent applies to $(-2)$ as a unit because of the parentheses, and the outer subtraction comes last.

The trap

Keeping the minus sign, (-2)^(-2) = -1/4, which turns the subtraction into 2 + 1/4 = 9/4, choice (D).

Common mistakes

Keeping the minus sign, $(-2)^{-2} = -\tfrac14$ , which turns the subtraction into $2 + \tfrac14 = \tfrac94$ , choice (D).
Reading $(-2)^{-2}$ as $-2^{-2}$ or as $(-2)^2 = 4$ and answering $-2$ , choice (A).

Techniques

Set up the equation/formula and compute; no special trick needed